Creat a positive definite matrix, which way is better?

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Ming 2016-1-12

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评论： Matt J 2016-1-12

Hi everyone,

Suppose I have a positive definite matrix P and a non-singular matrix A, I realised there is a small numerical difference between K=(I-A)*P*(I-A)' and K_=P-A*P-P*A'+A*P*A'. If I want to create a positive definite matrix, which way should I use?

In some extreme cases, is it possible that K or K_ has very small negative eigenvalues?

Thanks

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Matt J 2016-1-12

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编辑：Matt J 2016-1-12

Yes, if A has eigenvalues close to 1, then I believe neither K nor K_ will be positive definite with numerical robustness.

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Ming 2016-1-12

Thanks a lot! By the way, does this numerical problem/hazard have a formal name?

Matt J 2016-1-12

No, not that I know of.

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